Principally Polarized Abelian Schemes

• Principally polarized abelian schemes are smooth, proper group schemes equipped with a unique ample line bundle (theta divisor) that defines an isomorphism with their dual.
• They decompose uniquely into indecomposable factors as per Shimura's theorem, with irreducible theta divisors corresponding bijectively to each component.
• Polarization theory in these schemes interlinks Hermitian forms, isogeny compatibility, and moduli interpretations, influencing classification and automorphism analyses.

A principally polarized abelian scheme (PPAS) is a smooth, proper group scheme $\mathcal{A}\to S$ over a base scheme $S$, equipped with a principal polarization. This principal polarization is a relatively ample divisor $\Theta\subset\mathcal{A}$ such that, on each geometric fiber, the induced map to the dual abelian scheme is an isomorphism. The category of principally polarized abelian schemes encompasses foundational questions in algebraic geometry, arithmetic geometry, and moduli theory, including the unique factorization into indecomposables, characterization via theta divisors and secant conditions, classification of moduli spaces, and compatibility of isogenies with polarizations.

1. Definitions and Structural Properties

An abelian scheme $\mathcal{A}\to S$ is a proper, smooth group scheme whose geometric fibers are connected projective algebraic groups. Principal polarizations are given by symmetric ample line bundles $L_\lambda$ for which the adjunction morphism

$$\lambda: \mathcal{A} \longrightarrow \mathcal{A}^\vee,$$

is an isomorphism. The theta divisor $\Theta$ associated to $L_\lambda$ satisfies $[-1]_\mathcal{A}^*\Theta = \Theta$ and $h^0(\mathcal{A},L_\lambda)=1$. Over $\mathbb{C}$, PPAS correspond to complex tori equipped with an indivisible polarizing class.

Actions by rings $R$ (e.g., $\mathcal{O}_K$ for CM fields) provide additional structure; an $R$-linear polarization satisfies $\lambda\circ\iota(r) = \iota^\vee(r)\circ\lambda$ for $r\in R$ when $R$ is equipped with a positive involution.

2. Unique Factorization of Principally Polarized Abelian Schemes

Every PPAS admits a unique decomposition into indecomposable principally polarized abelian varieties. Shimura's theorem states that for a PPAV $(A,\lambda)$ over $\mathbb{C}$,

$(A,\lambda) \cong \prod_{i=1}^r (A_i,\lambda_i)$

where each $(A_i,\lambda_i)$ is indecomposable and the decomposition is unique up to order. This result generalizes to any separably closed field of characteristic $\ne2$, leveraging algebraic arguments via Tate modules and Dieudonné theory. The rational endomorphism algebra $\mathrm{End}^0(A)$ acquires a positive Rosati involution, splitting into simple factors corresponding to the indecomposable PPAVs.

Over a separably closed field, irreducible components of the theta divisor $\Theta$ bijectively correspond to indecomposable factors:

$$\{\text{indecomposable } A_i\} \longleftrightarrow \{\text{irreducible } \Theta_j\}.$$

For a base scheme $S$ with $2$ invertible, the decomposition, Rosati involution, and theta correspondence globalize étale-locally, yielding a unique splitting into indecomposable p.p. abelian $S$-schemes (Jordan et al., 2016).

3. Polarization Theory, Isogenies, and Hermitian Forms

Polarizations correspond to symmetric isogenies $\lambda : A \to A^{\vee}$ such that the associated line bundle is ample. Under suitable hypotheses, principal polarizations are characterized by nondegenerate hermitian forms on $R$-modules (for $R$-linear abelian schemes); Serre's tensor construction permits the construction of new abelian schemes from modules $M$ and the PPAS $(A,\iota,\lambda)$, with polarization $h\otimes \lambda$ if and only if $h$ is positive-definite Hermitian.

In endomorphism-theoretic terms, Albert's classification describes the types of division algebras $D$ with positive involution, and therefore the structure of polarizations as positive-definite Hermitian forms $\psi : V\times V\to D$, where $V$ is a right $D$-module.

Compatibility between isogenies and polarizations is governed algebraically. Given unpolarized isogenies $A \to B$ and polarizations $\lambda_A, \lambda_B$, one seeks $f^*\lambda_B = n\lambda_A$, i.e., isogenies compatible up to scalar. There is a uniform polynomial degree bound for polarised isogenies in terms of dimension and endomorphism data. Notably, the fourth-power theorem asserts the existence of a polarised isogeny between $(A,\lambda)^4$ and $(B,\mu)^4$ for any abelian varieties related by an unpolarized isogeny, via Hermitian form theory (Orr, 2015, Amir-Khosravi, 2015).

4. Principal Polarizations: Kernel Obstructions and Constructibility

The existence of principal polarizations in a given isogeny class of abelian varieties over finite fields is finely controlled by an explicit obstruction element $I_C$ in a finite 2-torsion group $B(R)$, which can be computed from the Frobenius characteristic polynomial. Under mild hypotheses, such as the abelian variety being of odd dimension or the corresponding field being totally real or having ramified primes in the CM case, the obstruction vanishes and a principal polarization exists.

For products of abelian varieties, the Serre "gluing" construction descends product polarizations to principal polarizations under precisely controlled anti-isometries of kernels; the gluing exponent $e(A_1,A_2)$ and resulting group schemes control the possibility of irreducible principal polarizations (Rybakov, 31 Mar 2024, Howe, 2020).

5. Moduli Spaces and Modular Interpretations

Moduli of PPAS, notably the stacks $M_\Phi^n$ for CM data $(K,\Phi)$, are algebro-geometrically classified. The Serre tensor construction establishes that any principally polarized $n$-dimensional CM abelian scheme is locally isomorphic to a tensor product $(M,h)\otimes(A_0,\iota_0,\lambda_0)$, with $(M,h)$ a Hermitian module and $(A_0,\iota_0,\lambda_0)$ in $M_\Phi^1$. The stack $M_\Phi^n$ is Deligne-Mumford, zero-dimensional, and coincides with integral models of zero-dimensional Shimura varieties associated to compact unitary groups (Amir-Khosravi, 2015).

6. Characterization Problems and Theta-Functional Geometry

The Schottky problem and its extensions seek to characterize Jacobians and Pryms among indecomposable PPAVs via geometric, modular, and soliton-theoretic criteria:

• The Welters–Krichever trisecant criterion states that a PPAV is a Jacobian if and only if its Kummer variety admits a trisecant, equivalently formulated as the existence of a solution of the KP auxiliary linear problem with pole ansatz form and theta-functional equations.
• The Prym locus is characterized by the existence of pairs of quadrisecants in the Kummer variety and corresponding theta identities.
• The Castelnuovo–Schottky theorem provides a scheme-theoretic criterion: a PPAV admitting a $\theta$-general subscheme of $g+2$ points imposing fewer than $g+2$ conditions with respect to $2\Theta$ is necessarily a Jacobian, and the subscheme lies on an Abel–Jacobi curve (Gulbrandsen et al., 2010, Krichever, 2022).

These geometric characterizations globalize to moduli stacks, cutting out loci of Jacobians, Pryms, and spectral curves with involution scheme-theoretically via explicit functional equations linking theta divisors, secant geometry, and integrable systems.

7. Automorphism Structure and Enumeration of Principal Polarizations

The number $\pi(A)$ of non-isomorphic principal polarizations on a PPAV $A$ is given by the number of orbits of Aut$(A)$ acting on principal polarizations. Concrete enumeration employs lattice-search techniques, Rosati-involution-fixed elements of End$(A)$, and positivity tests for Hermitian forms of determinant one. The methodology extends to cases with complex multiplication, hyperelliptic and cyclic cover Jacobians, and higher-dimensional factorization. These computations elucidate the automorphism group actions, provide lower bounds for $\pi(A)$, and contribute to deep conjectures regarding the landscape of principal polarizations across moduli spaces (Lee et al., 2018).

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Collectively, the theory of principally polarized abelian schemes integrates algebraic, geometric, and arithmetic perspectives, unified via polarization theory, endomorphism algebra, moduli classifications, and explicit geometric and functional criteria reflecting deep interconnections between abelian varieties, their moduli, and integrable systems.